Adding and subtracting whole numbers to a million
By the end of this chapter you will be able to
- Add two numbers to 999,999 in columns, carrying across several places
- Subtract in columns, exchanging across zeros
- Estimate by rounding, and check a sum or difference
- Find a missing number in a sum or difference
- Solve two-step problems with numbers to a million
You should already know
- Column addition and subtraction to 10,000 (Level 4, chapter 3)
- Place value to a million (chapter 1)
Where this leads
- Long multiplication (chapter 4)
- Fractions with the same denominator (chapter 6)
Step-by-step practice: maths (Level 5). Small steps, repeated until fluent. Content: Add and subtract whole numbers with more than four digits, using columnar methods (Y5 paraphrase), Use rounding to check answers and estimate, and decide the level of accuracy (Y5 paraphrase), Solve addition and subtraction multi-step problems (Y5 paraphrase), Solve missing number problems (Y5 paraphrase).
Exercise P Do you remember?
Non-calculatorMethod. Add the columns from the right. Carry any ten into the next column on the left.
- Add the ones column. Write 4 and carry 1 to the tens column.5 + 9 = 14
- Add the tens column. Write 9 in the tens column.7 + 1 + 1 carried = 9
- Add the hundreds column. Write 1 and carry 1 to the thousands column.3 + 8 = 11
- Add the thousands column. Write 7 in the thousands column.4 + 2 + 1 carried = 7
- Write the answer. Put the digits together, with commas between groups of three.7,194
Check. Check: 7,194 - 2,819 = 4,375.
Watch out. Carry the 1 into the next column. Do not forget it, and do not write it as a digit in its own column.
Method. Add the columns from the right. Carry any ten into the next column on the left.
- Add the ones column. Write 3 and carry 1 to the tens column.8 + 5 = 13
- Add the tens column. Write 1 and carry 1 to the hundreds column.4 + 6 + 1 carried = 11
- Add the hundreds column. Write 0 and carry 1 to the thousands column.0 + 9 + 1 carried = 10
- Add the thousands column. Write 8 in the thousands column.6 + 1 + 1 carried = 8
- Write the answer. Put the digits together, with commas between groups of three.8,013
Check. Check: 8,013 - 1,965 = 6,048.
Watch out. Carry the 1 into the next column. Do not forget it, and do not write it as a digit in its own column.
Method. Subtract each column from the right. Exchange 1 from the column on the left when the top digit is too small.
- Subtract the ones column. 0 is less than 6. The tens and hundreds are 0, so exchange across them: take 1 thousands and make each zero a 9, which gives 10 ones.10 - 6 = 4
- Subtract the tens column. 9 is at least 4, so subtract directly.9 - 4 = 5
- Subtract the hundreds column. 9 is at least 3, so subtract directly.9 - 3 = 6
- Subtract the thousands column. 4 is at least 2, so subtract directly.4 - 2 = 2
- Write the answer. Read the digits from the left, and put commas in.2,654
Check. Check: 2,346 + 2,654 = 5,000.
Watch out. When you exchange across zeros, change each zero to 9 as you pass it. Do not leave a zero in the middle.
Method. Use the 7 or 8 times table.
- Count in eights seven times. Each count adds 8.8, 16, 24, 32, 40, 48, 56
- The last number we said is the answer. We counted seven groups of 8.8 × 7 = 56
- Write the answer. This is the total.56
Check. Check: 56 ÷ 7 = 8.
Watch out. Count the eights carefully.
Method. Add 30,000 by adding 3 to the ten thousands digit.
- Look at the ten thousands digit. It is the second digit from the left in 245,000.The digit is 4
- Add 3 to it. Adding 30,000 changes the ten thousands digit by 3.4 + 3 = 7
- Write the answer. All the other digits stay the same.275,000
Check. Check: 275,000 - 30,000 = 245,000.
Watch out. Only the ten thousands digit changes.
Adding in columns with carrying
Line up the digits in columns, so that ones are under ones and thousands are under thousands. Start from the right.
When a column adds up to ten or more, write the ones digit and carry the ten to the next column on the left.
Key ideas
- Line up the columns, with the ones on the right.
- Add one column at a time, starting from the right.
- Carry a ten into the next column on the left.
Method. Add the columns from the right. Carry any ten into the next column on the left.
- Add the ones column. Write 7 in the ones column.4 + 3 = 7
- Add the tens column. Write 5 and carry 1 to the hundreds column.8 + 7 = 15
- Add the hundreds column. Write 3 and carry 1 to the thousands column.3 + 9 + 1 carried = 13
- Add the thousands column. Write 4 and carry 1 to the ten thousands column.5 + 8 + 1 carried = 14
- Add the ten thousands column. Write 1 and carry 1 to the hundred thousands column.6 + 4 + 1 carried = 11
- Add the hundred thousands column. Write 3 in the hundred thousands column.2 + 0 + 1 carried = 3
- Write the answer. Put the digits together, with commas between groups of three.314,357
Check. Check: 314,357 - 48,973 = 265,384.
Watch out. Carry the 1 into the next column. Do not forget it, and do not write it as a digit in its own column.
Method. Add the columns from the right. Carry any ten into the next column on the left.
- Add the ones column. Write 2 and carry 1 to the tens column.5 + 7 = 12
- Add the tens column. Write 1 and carry 1 to the hundreds column.6 + 4 + 1 carried = 11
- Add the hundreds column. Write 0 and carry 1 to the thousands column.7 + 2 + 1 carried = 10
- Add the thousands column. Write 0 and carry 1 to the ten thousands column.8 + 1 + 1 carried = 10
- Add the ten thousands column. Write 0 and carry 1 to the hundred thousands column.9 + 0 + 1 carried = 10
- Add the hundred thousands column. Write 0 and carry 1 to the millions column.6 + 3 + 1 carried = 10
- Write the last carried 1. It goes in front of the other digits.1 in the millions column
- Write the answer. Put the digits together, with commas between groups of three.1,000,012
Check. Check: 1,000,012 - 301,247 = 698,765.
Watch out. Carry the 1 into the next column. Do not forget it, and do not write it as a digit in its own column.
Method. Add the columns from the right. Carry any ten into the next column on the left.
- Add the ones column. Write 3 and carry 1 to the tens column.6 + 7 = 13
- Add the tens column. Write 2 and carry 1 to the hundreds column.5 + 6 + 1 carried = 12
- Add the hundreds column. Write 4 and carry 1 to the thousands column.4 + 9 + 1 carried = 14
- Add the thousands column. Write 6 and carry 1 to the ten thousands column.8 + 7 + 1 carried = 16
- Add the ten thousands column. Write 6 in the ten thousands column.3 + 2 + 1 carried = 6
- Write the answer. Put the digits together, with commas between groups of three.66,423
Check. Check: 66,423 - 27,967 = 38,456.
Watch out. Carry the 1 into the next column. Do not forget it, and do not write it as a digit in its own column.
Watch out
Watch out. Carried tens go into the next column, not the one under it. Add the carried number first.
Exam tip
Grown-up tip. Ask the child to say the carry aloud: “Seven and five is twelve, write two and carry one.”
Exercise 2A Practice: adding in columns
Non-calculatorMethod. Add the columns from the right. Carry any ten into the next column on the left.
- Add the ones column. Write 8 in the ones column.1 + 7 = 8
- Add the tens column. Write 1 and carry 1 to the hundreds column.8 + 3 = 11
- Add the hundreds column. Write 8 in the hundreds column.6 + 1 + 1 carried = 8
- Add the thousands column. Write 0 and carry 1 to the ten thousands column.2 + 8 = 10
- Add the ten thousands column. Write 4 and carry 1 to the hundred thousands column.7 + 6 + 1 carried = 14
- Add the hundred thousands column. Write 8 in the hundred thousands column.2 + 5 + 1 carried = 8
- Write the answer. Put the digits together, with commas between groups of three.840,818
Check. Check: 840,818 - 568,137 = 272,681.
Watch out. Carry the 1 into the next column. Do not forget it, and do not write it as a digit in its own column.
Method. Add the columns from the right. Carry any ten into the next column on the left.
- Add the ones column. Write 1 and carry 1 to the tens column.9 + 2 = 11
- Add the tens column. Write 6 and carry 1 to the hundreds column.9 + 6 + 1 carried = 16
- Add the hundreds column. Write 5 and carry 1 to the thousands column.9 + 5 + 1 carried = 15
- Add the thousands column. Write 1 and carry 1 to the ten thousands column.9 + 1 + 1 carried = 11
- Add the ten thousands column. Write 8 in the ten thousands column.2 + 5 + 1 carried = 8
- Add the hundred thousands column. Write 6 in the hundred thousands column.5 + 1 = 6
- Write the answer. Put the digits together, with commas between groups of three.681,561
Check. Check: 681,561 - 151,562 = 529,999.
Watch out. Carry the 1 into the next column. Do not forget it, and do not write it as a digit in its own column.
Method. Add the columns from the right. Carry any ten into the next column on the left.
- Add the ones column. Write 3 and carry 1 to the tens column.5 + 8 = 13
- Add the tens column. Write 7 and carry 1 to the hundreds column.7 + 9 + 1 carried = 17
- Add the hundreds column. Write 7 in the hundreds column.2 + 4 + 1 carried = 7
- Add the thousands column. Write 9 in the thousands column.1 + 8 = 9
- Add the ten thousands column. Write 6 and carry 1 to the hundred thousands column.8 + 8 = 16
- Add the hundred thousands column. Write 5 in the hundred thousands column.3 + 1 + 1 carried = 5
- Write the answer. Put the digits together, with commas between groups of three.569,773
Check. Check: 569,773 - 188,498 = 381,275.
Watch out. Carry the 1 into the next column. Do not forget it, and do not write it as a digit in its own column.
Method. Add the columns from the right. Carry any ten into the next column on the left.
- Add the ones column. Write 8 in the ones column.2 + 6 = 8
- Add the tens column. Write 6 in the tens column.1 + 5 = 6
- Add the hundreds column. Write 7 in the hundreds column.2 + 5 = 7
- Add the thousands column. Write 4 and carry 1 to the ten thousands column.8 + 6 = 14
- Add the ten thousands column. Write 2 and carry 1 to the hundred thousands column.2 + 9 + 1 carried = 12
- Add the hundred thousands column. Write 9 in the hundred thousands column.1 + 7 + 1 carried = 9
- Write the answer. Put the digits together, with commas between groups of three.924,768
Check. Check: 924,768 - 796,556 = 128,212.
Watch out. Carry the 1 into the next column. Do not forget it, and do not write it as a digit in its own column.
Method. Add the columns from the right. Carry any ten into the next column on the left.
- Add the ones column. Write 7 in the ones column.6 + 1 = 7
- Add the tens column. Write 7 in the tens column.2 + 5 = 7
- Add the hundreds column. Write 9 in the hundreds column.7 + 2 = 9
- Add the thousands column. Write 5 and carry 1 to the ten thousands column.6 + 9 = 15
- Add the ten thousands column. Write 6 in the ten thousands column.3 + 2 + 1 carried = 6
- Add the hundred thousands column. Write 9 in the hundred thousands column.8 + 1 = 9
- Write the answer. Put the digits together, with commas between groups of three.965,977
Check. Check: 965,977 - 129,251 = 836,726.
Watch out. Carry the 1 into the next column. Do not forget it, and do not write it as a digit in its own column.
Method. Add the columns from the right. Carry any ten into the next column on the left.
- Add the ones column. Write 1 and carry 1 to the tens column.5 + 6 = 11
- Add the tens column. Write 1 and carry 1 to the hundreds column.7 + 3 + 1 carried = 11
- Add the hundreds column. Write 6 in the hundreds column.5 + 0 + 1 carried = 6
- Add the thousands column. Write 5 and carry 1 to the ten thousands column.8 + 7 = 15
- Add the ten thousands column. Write 3 and carry 1 to the hundred thousands column.8 + 4 + 1 carried = 13
- Add the hundred thousands column. Write 9 in the hundred thousands column.8 + 0 + 1 carried = 9
- Write the answer. Put the digits together, with commas between groups of three.935,611
Check. Check: 935,611 - 47,036 = 888,575.
Watch out. Carry the 1 into the next column. Do not forget it, and do not write it as a digit in its own column.
Method. Add the columns from the right. Carry any ten into the next column on the left.
- Add the ones column. Write 8 in the ones column.8 + 0 = 8
- Add the tens column. Write 6 and carry 1 to the hundreds column.9 + 7 = 16
- Add the hundreds column. Write 4 in the hundreds column.2 + 1 + 1 carried = 4
- Add the thousands column. Write 9 in the thousands column.4 + 5 = 9
- Add the ten thousands column. Write 5 in the ten thousands column.5 + 0 = 5
- Add the hundred thousands column. Write 9 in the hundred thousands column.8 + 1 = 9
- Write the answer. Put the digits together, with commas between groups of three.959,468
Check. Check: 959,468 - 105,170 = 854,298.
Watch out. Carry the 1 into the next column. Do not forget it, and do not write it as a digit in its own column.
Method. Add the columns from the right. Carry any ten into the next column on the left.
- Add the ones column. Write 2 in the ones column.1 + 1 = 2
- Add the tens column. Write 6 and carry 1 to the hundreds column.8 + 8 = 16
- Add the hundreds column. Write 4 and carry 1 to the thousands column.6 + 7 + 1 carried = 14
- Add the thousands column. Write 3 and carry 1 to the ten thousands column.7 + 5 + 1 carried = 13
- Add the ten thousands column. Write 5 and carry 1 to the hundred thousands column.7 + 7 + 1 carried = 15
- Add the hundred thousands column. Write 6 in the hundred thousands column.3 + 2 + 1 carried = 6
- Write the answer. Put the digits together, with commas between groups of three.653,462
Check. Check: 653,462 - 275,781 = 377,681.
Watch out. Carry the 1 into the next column. Do not forget it, and do not write it as a digit in its own column.
Method. Add the columns from the right. Carry any ten into the next column on the left.
- Add the ones column. Write 1 and carry 1 to the tens column.7 + 4 = 11
- Add the tens column. Write 0 and carry 1 to the hundreds column.1 + 8 + 1 carried = 10
- Add the hundreds column. Write 2 and carry 1 to the thousands column.5 + 6 + 1 carried = 12
- Add the thousands column. Write 1 and carry 1 to the ten thousands column.6 + 4 + 1 carried = 11
- Add the ten thousands column. Write 5 in the ten thousands column.3 + 1 + 1 carried = 5
- Add the hundred thousands column. Write 5 in the hundred thousands column.1 + 4 = 5
- Write the answer. Put the digits together, with commas between groups of three.551,201
Check. Check: 551,201 - 414,684 = 136,517.
Watch out. Carry the 1 into the next column. Do not forget it, and do not write it as a digit in its own column.
Method. Add the columns from the right. Carry any ten into the next column on the left.
- Add the ones column. Write 0 and carry 1 to the tens column.9 + 1 = 10
- Add the tens column. Write 8 and carry 1 to the hundreds column.9 + 8 + 1 carried = 18
- Add the hundreds column. Write 5 and carry 1 to the thousands column.6 + 8 + 1 carried = 15
- Add the thousands column. Write 8 in the thousands column.0 + 7 + 1 carried = 8
- Add the ten thousands column. Write 7 in the ten thousands column.6 + 1 = 7
- Add the hundred thousands column. Write 3 in the hundred thousands column.0 + 3 = 3
- Write the answer. Put the digits together, with commas between groups of three.378,580
Check. Check: 378,580 - 317,881 = 60,699.
Watch out. Carry the 1 into the next column. Do not forget it, and do not write it as a digit in its own column.
Method. Add the columns from the right. Carry any ten into the next column on the left.
- Add the ones column. Write 2 and carry 1 to the tens column.6 + 6 = 12
- Add the tens column. Write 8 in the tens column.3 + 4 + 1 carried = 8
- Add the hundreds column. Write 1 and carry 1 to the thousands column.9 + 2 = 11
- Add the thousands column. Write 7 in the thousands column.0 + 6 + 1 carried = 7
- Add the ten thousands column. Write 7 in the ten thousands column.4 + 3 = 7
- Add the hundred thousands column. Write 4 in the hundred thousands column.2 + 2 = 4
- Write the answer. Put the digits together, with commas between groups of three.477,182
Check. Check: 477,182 - 236,246 = 240,936.
Watch out. Carry the 1 into the next column. Do not forget it, and do not write it as a digit in its own column.
Method. Add the columns from the right. Carry any ten into the next column on the left.
- Add the ones column. Write 7 in the ones column.0 + 7 = 7
- Add the tens column. Write 8 in the tens column.5 + 3 = 8
- Add the hundreds column. Write 0 and carry 1 to the thousands column.6 + 4 = 10
- Add the thousands column. Write 9 in the thousands column.2 + 6 + 1 carried = 9
- Add the ten thousands column. Write 0 and carry 1 to the hundred thousands column.4 + 6 = 10
- Add the hundred thousands column. Write 8 in the hundred thousands column.3 + 4 + 1 carried = 8
- Write the answer. Put the digits together, with commas between groups of three.809,087
Check. Check: 809,087 - 466,437 = 342,650.
Watch out. Carry the 1 into the next column. Do not forget it, and do not write it as a digit in its own column.
Method. Add the columns from the right. Carry any ten into the next column on the left.
- Add the ones column. Write 0 and carry 1 to the tens column.6 + 4 = 10
- Add the tens column. Write 8 in the tens column.7 + 0 + 1 carried = 8
- Add the hundreds column. Write 6 in the hundreds column.3 + 3 = 6
- Add the thousands column. Write 3 and carry 1 to the ten thousands column.5 + 8 = 13
- Add the ten thousands column. Write 9 in the ten thousands column.7 + 1 + 1 carried = 9
- Add the hundred thousands column. Write 9 in the hundred thousands column.8 + 1 = 9
- Write the answer. Put the digits together, with commas between groups of three.993,680
Check. Check: 993,680 - 118,304 = 875,376.
Watch out. Carry the 1 into the next column. Do not forget it, and do not write it as a digit in its own column.
Method. Add the columns from the right. Carry any ten into the next column on the left.
- Add the ones column. Write 0 and carry 1 to the tens column.1 + 9 = 10
- Add the tens column. Write 7 and carry 1 to the hundreds column.9 + 7 + 1 carried = 17
- Add the hundreds column. Write 2 and carry 1 to the thousands column.9 + 2 + 1 carried = 12
- Add the thousands column. Write 5 and carry 1 to the ten thousands column.9 + 5 + 1 carried = 15
- Add the ten thousands column. Write 6 in the ten thousands column.0 + 5 + 1 carried = 6
- Add the hundred thousands column. Write 6 in the hundred thousands column.5 + 1 = 6
- Write the answer. Put the digits together, with commas between groups of three.665,270
Check. Check: 665,270 - 155,279 = 509,991.
Watch out. Carry the 1 into the next column. Do not forget it, and do not write it as a digit in its own column.
Method. Add the columns from the right. Carry any ten into the next column on the left.
- Add the ones column. Write 9 in the ones column.6 + 3 = 9
- Add the tens column. Write 9 in the tens column.4 + 5 = 9
- Add the hundreds column. Write 6 in the hundreds column.6 + 0 = 6
- Add the thousands column. Write 7 in the thousands column.7 + 0 = 7
- Add the ten thousands column. Write 6 in the ten thousands column.4 + 2 = 6
- Add the hundred thousands column. Write 6 in the hundred thousands column.2 + 4 = 6
- Write the answer. Put the digits together, with commas between groups of three.667,699
Check. Check: 667,699 - 420,053 = 247,646.
Watch out. Carry the 1 into the next column. Do not forget it, and do not write it as a digit in its own column.
Method. Add the columns from the right. Carry any ten into the next column on the left.
- Add the ones column. Write 4 in the ones column.2 + 2 = 4
- Add the tens column. Write 6 in the tens column.4 + 2 = 6
- Add the hundreds column. Write 3 in the hundreds column.3 + 0 = 3
- Add the thousands column. Write 0 and carry 1 to the ten thousands column.5 + 5 = 10
- Add the ten thousands column. Write 3 in the ten thousands column.1 + 1 + 1 carried = 3
- Add the hundred thousands column. Write 7 in the hundred thousands column.5 + 2 = 7
- Write the answer. Put the digits together, with commas between groups of three.730,364
Check. Check: 730,364 - 215,022 = 515,342.
Watch out. Carry the 1 into the next column. Do not forget it, and do not write it as a digit in its own column.
Carrying across several places
Some sums carry in every column. Each carried 1 is added to the next column, so it can make another ten.
Check by subtracting one of the numbers from the total. You should get the other number back.
Key ideas
- Carries can happen in several columns in one sum.
- Add the carried 1 before you add the next pair of digits.
- Check the total by taking away one of the numbers.
Method. Add the columns from the right. Carry any ten into the next column on the left.
- Add the ones column. Write 1 and carry 1 to the tens column.9 + 2 = 11
- Add the tens column. Write 4 and carry 1 to the hundreds column.4 + 9 + 1 carried = 14
- Add the hundreds column. Write 4 and carry 1 to the thousands column.5 + 8 + 1 carried = 14
- Add the thousands column. Write 4 and carry 1 to the ten thousands column.6 + 7 + 1 carried = 14
- Add the ten thousands column. Write 5 and carry 1 to the hundred thousands column.8 + 6 + 1 carried = 15
- Add the hundred thousands column. Write 8 in the hundred thousands column.2 + 5 + 1 carried = 8
- Write the answer. Put the digits together, with commas between groups of three.854,441
Check. Check: 854,441 - 567,892 = 286,549.
Watch out. Carry the 1 into the next column. Do not forget it, and do not write it as a digit in its own column.
Method. Check by using the inverse: subtract for a sum, or add back for a difference.
- Take one number away from the answer. If the sum is correct, taking one number away gives the other.700,001 - 582,463 = 117,538
- Compare with the other number. Is 117,538 the same as 117,538?Same
- Write Yes or No. Yes if the check works; No if it does not.Yes
Check. The correct answer is 700,001.
Watch out. A check that does not work means the answer is wrong. Find the mistake in the columns.
Method. Add the columns from the right. Carry any ten into the next column on the left.
- Add the ones column. Write 2 and carry 1 to the tens column.5 + 7 = 12
- Add the tens column. Write 1 and carry 1 to the hundreds column.6 + 4 + 1 carried = 11
- Add the hundreds column. Write 0 and carry 1 to the thousands column.7 + 2 + 1 carried = 10
- Add the thousands column. Write 0 and carry 1 to the ten thousands column.8 + 1 + 1 carried = 10
- Add the ten thousands column. Write 0 and carry 1 to the hundred thousands column.9 + 0 + 1 carried = 10
- Add the hundred thousands column. Write 0 and carry 1 to the millions column.6 + 3 + 1 carried = 10
- Write the last carried 1. It goes in front of the other digits.1 in the millions column
- Write the answer. Put the digits together, with commas between groups of three.1,000,012
Check. Check: 1,000,012 - 301,247 = 698,765.
Watch out. Carry the 1 into the next column. Do not forget it, and do not write it as a digit in its own column.
Watch out
Watch out. A total of 1,000,000 or more has seven digits. Make sure the answer has the right number of digits.
Exam tip
Grown-up tip. Ask the child to check every big sum by subtracting one number from the total.
Exercise 2B Practice: carrying across several places
Non-calculatorMethod. Add the columns from the right. Carry any ten into the next column on the left.
- Add the ones column. Write 3 and carry 1 to the tens column.6 + 7 = 13
- Add the tens column. Write 1 and carry 1 to the hundreds column.7 + 3 + 1 carried = 11
- Add the hundreds column. Write 7 and carry 1 to the thousands column.9 + 7 + 1 carried = 17
- Add the thousands column. Write 3 in the thousands column.2 + 0 + 1 carried = 3
- Add the ten thousands column. Write 6 in the ten thousands column.0 + 6 = 6
- Add the hundred thousands column. Write 6 in the hundred thousands column.4 + 2 = 6
- Write the answer. Put the digits together, with commas between groups of three.663,713
Check. Check: 663,713 - 260,737 = 402,976.
Watch out. Carry the 1 into the next column. Do not forget it, and do not write it as a digit in its own column.
Method. Add the columns from the right. Carry any ten into the next column on the left.
- Add the ones column. Write 0 and carry 1 to the tens column.5 + 5 = 10
- Add the tens column. Write 2 in the tens column.1 + 0 + 1 carried = 2
- Add the hundreds column. Write 5 and carry 1 to the thousands column.7 + 8 = 15
- Add the thousands column. Write 8 in the thousands column.2 + 5 + 1 carried = 8
- Add the ten thousands column. Write 8 in the ten thousands column.0 + 8 = 8
- Add the hundred thousands column. Write 8 in the hundred thousands column.8 + 0 = 8
- Write the answer. Put the digits together, with commas between groups of three.888,520
Check. Check: 888,520 - 85,805 = 802,715.
Watch out. Carry the 1 into the next column. Do not forget it, and do not write it as a digit in its own column.
Method. Add the columns from the right. Carry any ten into the next column on the left.
- Add the ones column. Write 7 and carry 1 to the tens column.8 + 9 = 17
- Add the tens column. Write 9 in the tens column.2 + 6 + 1 carried = 9
- Add the hundreds column. Write 0 and carry 1 to the thousands column.4 + 6 = 10
- Add the thousands column. Write 1 in the thousands column.0 + 0 + 1 carried = 1
- Add the ten thousands column. Write 3 and carry 1 to the hundred thousands column.5 + 8 = 13
- Add the hundred thousands column. Write 3 in the hundred thousands column.0 + 2 + 1 carried = 3
- Write the answer. Put the digits together, with commas between groups of three.331,097
Check. Check: 331,097 - 280,669 = 50,428.
Watch out. Carry the 1 into the next column. Do not forget it, and do not write it as a digit in its own column.
Method. Add the columns from the right. Carry any ten into the next column on the left.
- Add the ones column. Write 5 and carry 1 to the tens column.9 + 6 = 15
- Add the tens column. Write 9 in the tens column.6 + 2 + 1 carried = 9
- Add the hundreds column. Write 8 in the hundreds column.1 + 7 = 8
- Add the thousands column. Write 5 and carry 1 to the ten thousands column.6 + 9 = 15
- Add the ten thousands column. Write 9 in the ten thousands column.7 + 1 + 1 carried = 9
- Add the hundred thousands column. Write 9 in the hundred thousands column.8 + 1 = 9
- Write the answer. Put the digits together, with commas between groups of three.995,895
Check. Check: 995,895 - 119,726 = 876,169.
Watch out. Carry the 1 into the next column. Do not forget it, and do not write it as a digit in its own column.
Method. Add the columns from the right. Carry any ten into the next column on the left.
- Add the ones column. Write 0 and carry 1 to the tens column.3 + 7 = 10
- Add the tens column. Write 4 and carry 1 to the hundreds column.6 + 7 + 1 carried = 14
- Add the hundreds column. Write 9 in the hundreds column.4 + 4 + 1 carried = 9
- Add the thousands column. Write 1 and carry 1 to the ten thousands column.7 + 4 = 11
- Add the ten thousands column. Write 4 and carry 1 to the hundred thousands column.6 + 7 + 1 carried = 14
- Add the hundred thousands column. Write 8 in the hundred thousands column.7 + 0 + 1 carried = 8
- Write the answer. Put the digits together, with commas between groups of three.841,940
Check. Check: 841,940 - 74,477 = 767,463.
Watch out. Carry the 1 into the next column. Do not forget it, and do not write it as a digit in its own column.
Method. Add the columns from the right. Carry any ten into the next column on the left.
- Add the ones column. Write 0 and carry 1 to the tens column.8 + 2 = 10
- Add the tens column. Write 8 in the tens column.5 + 2 + 1 carried = 8
- Add the hundreds column. Write 5 and carry 1 to the thousands column.6 + 9 = 15
- Add the thousands column. Write 3 and carry 1 to the ten thousands column.6 + 6 + 1 carried = 13
- Add the ten thousands column. Write 4 in the ten thousands column.1 + 2 + 1 carried = 4
- Add the hundred thousands column. Write 4 in the hundred thousands column.1 + 3 = 4
- Write the answer. Put the digits together, with commas between groups of three.443,580
Check. Check: 443,580 - 326,922 = 116,658.
Watch out. Carry the 1 into the next column. Do not forget it, and do not write it as a digit in its own column.
Method. Add the columns from the right. Carry any ten into the next column on the left.
- Add the ones column. Write 6 in the ones column.4 + 2 = 6
- Add the tens column. Write 8 in the tens column.2 + 6 = 8
- Add the hundreds column. Write 1 and carry 1 to the thousands column.7 + 4 = 11
- Add the thousands column. Write 2 and carry 1 to the ten thousands column.3 + 8 + 1 carried = 12
- Add the ten thousands column. Write 6 in the ten thousands column.3 + 2 + 1 carried = 6
- Add the hundred thousands column. Write 5 in the hundred thousands column.5 + 0 = 5
- Write the answer. Put the digits together, with commas between groups of three.562,186
Check. Check: 562,186 - 28,462 = 533,724.
Watch out. Carry the 1 into the next column. Do not forget it, and do not write it as a digit in its own column.
Method. Add the columns from the right. Carry any ten into the next column on the left.
- Add the ones column. Write 9 in the ones column.6 + 3 = 9
- Add the tens column. Write 4 in the tens column.2 + 2 = 4
- Add the hundreds column. Write 0 and carry 1 to the thousands column.5 + 5 = 10
- Add the thousands column. Write 0 and carry 1 to the ten thousands column.9 + 0 + 1 carried = 10
- Add the ten thousands column. Write 8 in the ten thousands column.3 + 4 + 1 carried = 8
- Add the hundred thousands column. Write 6 in the hundred thousands column.2 + 4 = 6
- Write the answer. Put the digits together, with commas between groups of three.680,049
Check. Check: 680,049 - 440,523 = 239,526.
Watch out. Carry the 1 into the next column. Do not forget it, and do not write it as a digit in its own column.
Method. Add the columns from the right. Carry any ten into the next column on the left.
- Add the ones column. Write 0 and carry 1 to the tens column.7 + 3 = 10
- Add the tens column. Write 7 in the tens column.2 + 4 + 1 carried = 7
- Add the hundreds column. Write 3 and carry 1 to the thousands column.7 + 6 = 13
- Add the thousands column. Write 9 in the thousands column.8 + 0 + 1 carried = 9
- Add the ten thousands column. Write 6 in the ten thousands column.6 + 0 = 6
- Add the hundred thousands column. Write 5 in the hundred thousands column.2 + 3 = 5
- Write the answer. Put the digits together, with commas between groups of three.569,370
Check. Check: 569,370 - 300,643 = 268,727.
Watch out. Carry the 1 into the next column. Do not forget it, and do not write it as a digit in its own column.
Method. Add the columns from the right. Carry any ten into the next column on the left.
- Add the ones column. Write 6 in the ones column.1 + 5 = 6
- Add the tens column. Write 0 and carry 1 to the hundreds column.7 + 3 = 10
- Add the hundreds column. Write 3 and carry 1 to the thousands column.4 + 8 + 1 carried = 13
- Add the thousands column. Write 4 and carry 1 to the ten thousands column.4 + 9 + 1 carried = 14
- Add the ten thousands column. Write 2 in the ten thousands column.0 + 1 + 1 carried = 2
- Add the hundred thousands column. Write 6 in the hundred thousands column.6 + 0 = 6
- Write the answer. Put the digits together, with commas between groups of three.624,306
Check. Check: 624,306 - 19,835 = 604,471.
Watch out. Carry the 1 into the next column. Do not forget it, and do not write it as a digit in its own column.
Method. Add the columns from the right. Carry any ten into the next column on the left.
- Add the ones column. Write 5 in the ones column.2 + 3 = 5
- Add the tens column. Write 3 in the tens column.0 + 3 = 3
- Add the hundreds column. Write 2 and carry 1 to the thousands column.6 + 6 = 12
- Add the thousands column. Write 2 and carry 1 to the ten thousands column.2 + 9 + 1 carried = 12
- Add the ten thousands column. Write 9 in the ten thousands column.5 + 3 + 1 carried = 9
- Add the hundred thousands column. Write 8 in the hundred thousands column.5 + 3 = 8
- Write the answer. Put the digits together, with commas between groups of three.892,235
Check. Check: 892,235 - 339,633 = 552,602.
Watch out. Carry the 1 into the next column. Do not forget it, and do not write it as a digit in its own column.
Method. Add the columns from the right. Carry any ten into the next column on the left.
- Add the ones column. Write 7 and carry 1 to the tens column.9 + 8 = 17
- Add the tens column. Write 9 in the tens column.8 + 0 + 1 carried = 9
- Add the hundreds column. Write 9 in the hundreds column.6 + 3 = 9
- Add the thousands column. Write 7 in the thousands column.6 + 1 = 7
- Add the ten thousands column. Write 1 and carry 1 to the hundred thousands column.3 + 8 = 11
- Add the hundred thousands column. Write 9 in the hundred thousands column.5 + 3 + 1 carried = 9
- Write the answer. Put the digits together, with commas between groups of three.917,997
Check. Check: 917,997 - 381,308 = 536,689.
Watch out. Carry the 1 into the next column. Do not forget it, and do not write it as a digit in its own column.
Method. Add the columns from the right. Carry any ten into the next column on the left.
- Add the ones column. Write 3 and carry 1 to the tens column.6 + 7 = 13
- Add the tens column. Write 1 and carry 1 to the hundreds column.1 + 9 + 1 carried = 11
- Add the hundreds column. Write 3 and carry 1 to the thousands column.5 + 7 + 1 carried = 13
- Add the thousands column. Write 7 in the thousands column.6 + 0 + 1 carried = 7
- Add the ten thousands column. Write 4 in the ten thousands column.3 + 1 = 4
- Add the hundred thousands column. Write 8 in the hundred thousands column.3 + 5 = 8
- Write the answer. Put the digits together, with commas between groups of three.847,313
Check. Check: 847,313 - 510,797 = 336,516.
Watch out. Carry the 1 into the next column. Do not forget it, and do not write it as a digit in its own column.
Method. Add the columns from the right. Carry any ten into the next column on the left.
- Add the ones column. Write 4 and carry 1 to the tens column.9 + 5 = 14
- Add the tens column. Write 4 and carry 1 to the hundreds column.5 + 8 + 1 carried = 14
- Add the hundreds column. Write 8 in the hundreds column.6 + 1 + 1 carried = 8
- Add the thousands column. Write 5 in the thousands column.5 + 0 = 5
- Add the ten thousands column. Write 3 in the ten thousands column.0 + 3 = 3
- Add the hundred thousands column. Write 9 in the hundred thousands column.8 + 1 = 9
- Write the answer. Put the digits together, with commas between groups of three.935,844
Check. Check: 935,844 - 130,185 = 805,659.
Watch out. Carry the 1 into the next column. Do not forget it, and do not write it as a digit in its own column.
Method. Add the columns from the right. Carry any ten into the next column on the left.
- Add the ones column. Write 8 in the ones column.7 + 1 = 8
- Add the tens column. Write 0 and carry 1 to the hundreds column.4 + 6 = 10
- Add the hundreds column. Write 2 and carry 1 to the thousands column.9 + 2 + 1 carried = 12
- Add the thousands column. Write 7 in the thousands column.4 + 2 + 1 carried = 7
- Add the ten thousands column. Write 1 and carry 1 to the hundred thousands column.3 + 8 = 11
- Add the hundred thousands column. Write 9 in the hundred thousands column.5 + 3 + 1 carried = 9
- Write the answer. Put the digits together, with commas between groups of three.917,208
Check. Check: 917,208 - 382,261 = 534,947.
Watch out. Carry the 1 into the next column. Do not forget it, and do not write it as a digit in its own column.
Method. Add the columns from the right. Carry any ten into the next column on the left.
- Add the ones column. Write 1 and carry 1 to the tens column.8 + 3 = 11
- Add the tens column. Write 1 in the tens column.0 + 0 + 1 carried = 1
- Add the hundreds column. Write 9 in the hundreds column.5 + 4 = 9
- Add the thousands column. Write 1 and carry 1 to the ten thousands column.3 + 8 = 11
- Add the ten thousands column. Write 5 in the ten thousands column.2 + 2 + 1 carried = 5
- Add the hundred thousands column. Write 7 in the hundred thousands column.7 + 0 = 7
- Write the answer. Put the digits together, with commas between groups of three.751,911
Check. Check: 751,911 - 28,403 = 723,508.
Watch out. Carry the 1 into the next column. Do not forget it, and do not write it as a digit in its own column.
Subtracting in columns, with exchange
When the digit on top is smaller than the digit below, exchange 1 from the column on the left. That 1 is worth 10 in the column you are working in.
If the column on the left is 0, exchange across it. Each zero becomes a 9 as you pass it, and the next digit that is not zero gives 1.
Key ideas
- Exchange when the top digit is smaller.
- Exchange across zeros: a zero becomes 9 as you pass it.
- Work from the right, one column at a time.
Method. Subtract each column from the right. Exchange 1 from the column on the left when the top digit is too small.
- Subtract the ones column. 0 is less than 8. The tens and hundreds and thousands and ten thousands are 0, so exchange across them: take 1 hundred thousands and make each zero a 9, which gives 10 ones.10 - 8 = 2
- Subtract the tens column. 9 is at least 6, so subtract directly.9 - 6 = 3
- Subtract the hundreds column. 9 is at least 4, so subtract directly.9 - 4 = 5
- Subtract the thousands column. 9 is at least 3, so subtract directly.9 - 3 = 6
- Subtract the ten thousands column. 9 is at least 7, so subtract directly.9 - 7 = 2
- Subtract the hundred thousands column. 4 is at least 2, so subtract directly.4 - 2 = 2
- Write the answer. Read the digits from the left, and put commas in.226,532
Check. Check: 273,468 + 226,532 = 500,000.
Watch out. When you exchange across zeros, change each zero to 9 as you pass it. Do not leave a zero in the middle.
Method. Subtract each column from the right. Exchange 1 from the column on the left when the top digit is too small.
- Subtract the ones column. 5 is less than 6. The tens is 0, so exchange across them: take 1 hundreds and make each zero a 9, which gives 10 ones.15 - 6 = 9
- Subtract the tens column. 9 is at least 3, so subtract directly.9 - 3 = 6
- Subtract the hundreds column. 6 is at least 2, so subtract directly.6 - 2 = 4
- Subtract the thousands column. 0 is less than 8. The ten thousands is 0, so exchange across them: take 1 hundred thousands and make each zero a 9, which gives 10 thousands.10 - 8 = 2
- Subtract the ten thousands column. 9 is at least 9, so subtract directly.9 - 9 = 0
- Subtract the hundred thousands column. 3 is at least 0, so subtract directly.3 - 0 = 3
- Write the answer. Read the digits from the left, and put commas in.302,469
Check. Check: 98,236 + 302,469 = 400,705.
Watch out. When you exchange across zeros, change each zero to 9 as you pass it. Do not leave a zero in the middle.
Method. Subtract each column from the right. Exchange 1 from the column on the left when the top digit is too small.
- Subtract the ones column. 0 is less than 6, so exchange 1 tens for 10 ones.10 - 6 = 4
- Subtract the tens column. 2 is less than 9, so exchange 1 hundreds for 10 tens.12 - 9 = 3
- Subtract the hundreds column. 4 is at least 0, so subtract directly.4 - 0 = 4
- Subtract the thousands column. 2 is less than 8. The ten thousands is 0, so exchange across them: take 1 hundred thousands and make each zero a 9, which gives 10 thousands.12 - 8 = 4
- Subtract the ten thousands column. 9 is at least 1, so subtract directly.9 - 1 = 8
- Subtract the hundred thousands column. 6 is at least 4, so subtract directly.6 - 4 = 2
- Write the answer. Read the digits from the left, and put commas in.284,434
Check. Check: 418,096 + 284,434 = 702,530.
Watch out. When you exchange across zeros, change each zero to 9 as you pass it. Do not leave a zero in the middle.
Watch out
Watch out. In 500,000 - 273,468 the top number has zeros in the middle. Every zero must be exchanged across, one column at a time.
Exam tip
Grown-up tip. Ask: “Is the top digit smaller? Which column do you exchange from?” before the child starts to take away.
Exercise 2C Practice: subtracting with exchange
Non-calculatorMethod. Subtract each column from the right. Exchange 1 from the column on the left when the top digit is too small.
- Subtract the ones column. 9 is at least 9, so subtract directly.9 - 9 = 0
- Subtract the tens column. 2 is less than 8, so exchange 1 hundreds for 10 tens.12 - 8 = 4
- Subtract the hundreds column. 1 is less than 2, so exchange 1 thousands for 10 hundreds.11 - 2 = 9
- Subtract the thousands column. 8 is at least 0, so subtract directly.8 - 0 = 8
- Subtract the ten thousands column. 1 is less than 4, so exchange 1 hundred thousands for 10 ten thousands.11 - 4 = 7
- Subtract the hundred thousands column. 5 is at least 1, so subtract directly.5 - 1 = 4
- Write the answer. Read the digits from the left, and put commas in.478,940
Check. Check: 140,289 + 478,940 = 619,229.
Watch out. When you exchange across zeros, change each zero to 9 as you pass it. Do not leave a zero in the middle.
Method. Subtract each column from the right. Exchange 1 from the column on the left when the top digit is too small.
- Subtract the ones column. 7 is less than 8, so exchange 1 tens for 10 ones.17 - 8 = 9
- Subtract the tens column. 2 is at least 1, so subtract directly.2 - 1 = 1
- Subtract the hundreds column. 1 is less than 7, so exchange 1 thousands for 10 hundreds.11 - 7 = 4
- Subtract the thousands column. 3 is less than 6. The ten thousands is 0, so exchange across them: take 1 hundred thousands and make each zero a 9, which gives 10 thousands.13 - 6 = 7
- Subtract the ten thousands column. 9 is at least 5, so subtract directly.9 - 5 = 4
- Subtract the hundred thousands column. 3 is at least 3, so subtract directly.3 - 3 = 0
- Write the answer. Read the digits from the left, and put commas in.47,419
Check. Check: 356,718 + 47,419 = 404,137.
Watch out. When you exchange across zeros, change each zero to 9 as you pass it. Do not leave a zero in the middle.
Method. Subtract each column from the right. Exchange 1 from the column on the left when the top digit is too small.
- Subtract the ones column. 5 is less than 6, so exchange 1 tens for 10 ones.15 - 6 = 9
- Subtract the tens column. 3 is less than 8. The hundreds is 0, so exchange across them: take 1 thousands and make each zero a 9, which gives 10 tens.13 - 8 = 5
- Subtract the hundreds column. 9 is at least 5, so subtract directly.9 - 5 = 4
- Subtract the thousands column. 5 is less than 7, so exchange 1 ten thousands for 10 thousands.15 - 7 = 8
- Subtract the ten thousands column. 7 is less than 9, so exchange 1 hundred thousands for 10 ten thousands.17 - 9 = 8
- Subtract the hundred thousands column. 6 is at least 5, so subtract directly.6 - 5 = 1
- Write the answer. Read the digits from the left, and put commas in.188,459
Check. Check: 597,586 + 188,459 = 786,045.
Watch out. When you exchange across zeros, change each zero to 9 as you pass it. Do not leave a zero in the middle.
Method. Subtract each column from the right. Exchange 1 from the column on the left when the top digit is too small.
- Subtract the ones column. 1 is less than 8, so exchange 1 tens for 10 ones.11 - 8 = 3
- Subtract the tens column. 2 is at least 1, so subtract directly.2 - 1 = 1
- Subtract the hundreds column. 2 is less than 8, so exchange 1 thousands for 10 hundreds.12 - 8 = 4
- Subtract the thousands column. 2 is less than 9, so exchange 1 ten thousands for 10 thousands.12 - 9 = 3
- Subtract the ten thousands column. 3 is at least 3, so subtract directly.3 - 3 = 0
- Subtract the hundred thousands column. 6 is at least 1, so subtract directly.6 - 1 = 5
- Write the answer. Read the digits from the left, and put commas in.503,413
Check. Check: 139,818 + 503,413 = 643,231.
Watch out. When you exchange across zeros, change each zero to 9 as you pass it. Do not leave a zero in the middle.
Method. Subtract each column from the right. Exchange 1 from the column on the left when the top digit is too small.
- Subtract the ones column. 6 is at least 6, so subtract directly.6 - 6 = 0
- Subtract the tens column. 0 is less than 2, so exchange 1 hundreds for 10 tens.10 - 2 = 8
- Subtract the hundreds column. 2 is less than 3, so exchange 1 thousands for 10 hundreds.12 - 3 = 9
- Subtract the thousands column. 6 is at least 0, so subtract directly.6 - 0 = 6
- Subtract the ten thousands column. 7 is less than 8, so exchange 1 hundred thousands for 10 ten thousands.17 - 8 = 9
- Subtract the hundred thousands column. 3 is at least 3, so subtract directly.3 - 3 = 0
- Write the answer. Read the digits from the left, and put commas in.96,980
Check. Check: 380,326 + 96,980 = 477,306.
Watch out. When you exchange across zeros, change each zero to 9 as you pass it. Do not leave a zero in the middle.
Method. Subtract each column from the right. Exchange 1 from the column on the left when the top digit is too small.
- Subtract the ones column. 7 is at least 3, so subtract directly.7 - 3 = 4
- Subtract the tens column. 2 is less than 8, so exchange 1 hundreds for 10 tens.12 - 8 = 4
- Subtract the hundreds column. 4 is at least 2, so subtract directly.4 - 2 = 2
- Subtract the thousands column. 9 is at least 2, so subtract directly.9 - 2 = 7
- Subtract the ten thousands column. 4 is less than 5, so exchange 1 hundred thousands for 10 ten thousands.14 - 5 = 9
- Subtract the hundred thousands column. 8 is at least 0, so subtract directly.8 - 0 = 8
- Write the answer. Read the digits from the left, and put commas in.897,244
Check. Check: 52,283 + 897,244 = 949,527.
Watch out. When you exchange across zeros, change each zero to 9 as you pass it. Do not leave a zero in the middle.
Method. Subtract each column from the right. Exchange 1 from the column on the left when the top digit is too small.
- Subtract the ones column. 3 is at least 1, so subtract directly.3 - 1 = 2
- Subtract the tens column. 7 is at least 1, so subtract directly.7 - 1 = 6
- Subtract the hundreds column. 5 is less than 6, so exchange 1 thousands for 10 hundreds.15 - 6 = 9
- Subtract the thousands column. 8 is at least 7, so subtract directly.8 - 7 = 1
- Subtract the ten thousands column. 9 is at least 5, so subtract directly.9 - 5 = 4
- Subtract the hundred thousands column. 7 is at least 6, so subtract directly.7 - 6 = 1
- Write the answer. Read the digits from the left, and put commas in.141,962
Check. Check: 657,611 + 141,962 = 799,573.
Watch out. When you exchange across zeros, change each zero to 9 as you pass it. Do not leave a zero in the middle.
Method. Subtract each column from the right. Exchange 1 from the column on the left when the top digit is too small.
- Subtract the ones column. 9 is at least 4, so subtract directly.9 - 4 = 5
- Subtract the tens column. 7 is at least 6, so subtract directly.7 - 6 = 1
- Subtract the hundreds column. 0 is less than 4, so exchange 1 thousands for 10 hundreds.10 - 4 = 6
- Subtract the thousands column. 7 is at least 5, so subtract directly.7 - 5 = 2
- Subtract the ten thousands column. 5 is less than 6, so exchange 1 hundred thousands for 10 ten thousands.15 - 6 = 9
- Subtract the hundred thousands column. 3 is at least 0, so subtract directly.3 - 0 = 3
- Write the answer. Read the digits from the left, and put commas in.392,615
Check. Check: 65,464 + 392,615 = 458,079.
Watch out. When you exchange across zeros, change each zero to 9 as you pass it. Do not leave a zero in the middle.
Method. Subtract each column from the right. Exchange 1 from the column on the left when the top digit is too small.
- Subtract the ones column. 7 is at least 4, so subtract directly.7 - 4 = 3
- Subtract the tens column. 9 is at least 7, so subtract directly.9 - 7 = 2
- Subtract the hundreds column. 2 is less than 9. The thousands is 0, so exchange across them: take 1 ten thousands and make each zero a 9, which gives 10 hundreds.12 - 9 = 3
- Subtract the thousands column. 9 is at least 2, so subtract directly.9 - 2 = 7
- Subtract the ten thousands column. 2 is less than 7, so exchange 1 hundred thousands for 10 ten thousands.12 - 7 = 5
- Subtract the hundred thousands column. 6 is at least 0, so subtract directly.6 - 0 = 6
- Write the answer. Read the digits from the left, and put commas in.657,323
Check. Check: 72,974 + 657,323 = 730,297.
Watch out. When you exchange across zeros, change each zero to 9 as you pass it. Do not leave a zero in the middle.
Method. Subtract each column from the right. Exchange 1 from the column on the left when the top digit is too small.
- Subtract the ones column. 9 is at least 6, so subtract directly.9 - 6 = 3
- Subtract the tens column. 1 is less than 7, so exchange 1 hundreds for 10 tens.11 - 7 = 4
- Subtract the hundreds column. 7 is at least 2, so subtract directly.7 - 2 = 5
- Subtract the thousands column. 3 is less than 5, so exchange 1 ten thousands for 10 thousands.13 - 5 = 8
- Subtract the ten thousands column. 6 is at least 6, so subtract directly.6 - 6 = 0
- Subtract the hundred thousands column. 2 is at least 1, so subtract directly.2 - 1 = 1
- Write the answer. Read the digits from the left, and put commas in.108,543
Check. Check: 165,276 + 108,543 = 273,819.
Watch out. When you exchange across zeros, change each zero to 9 as you pass it. Do not leave a zero in the middle.
Method. Subtract each column from the right. Exchange 1 from the column on the left when the top digit is too small.
- Subtract the ones column. 4 is at least 3, so subtract directly.4 - 3 = 1
- Subtract the tens column. 7 is at least 7, so subtract directly.7 - 7 = 0
- Subtract the hundreds column. 5 is at least 0, so subtract directly.5 - 0 = 5
- Subtract the thousands column. 5 is less than 6, so exchange 1 ten thousands for 10 thousands.15 - 6 = 9
- Subtract the ten thousands column. 0 is less than 1, so exchange 1 hundred thousands for 10 ten thousands.10 - 1 = 9
- Subtract the hundred thousands column. 5 is at least 5, so subtract directly.5 - 5 = 0
- Write the answer. Read the digits from the left, and put commas in.99,501
Check. Check: 516,073 + 99,501 = 615,574.
Watch out. When you exchange across zeros, change each zero to 9 as you pass it. Do not leave a zero in the middle.
Method. Subtract each column from the right. Exchange 1 from the column on the left when the top digit is too small.
- Subtract the ones column. 7 is at least 0, so subtract directly.7 - 0 = 7
- Subtract the tens column. 4 is less than 6, so exchange 1 hundreds for 10 tens.14 - 6 = 8
- Subtract the hundreds column. 3 is less than 8, so exchange 1 thousands for 10 hundreds.13 - 8 = 5
- Subtract the thousands column. 5 is less than 8, so exchange 1 ten thousands for 10 thousands.15 - 8 = 7
- Subtract the ten thousands column. 7 is at least 6, so subtract directly.7 - 6 = 1
- Subtract the hundred thousands column. 6 is at least 2, so subtract directly.6 - 2 = 4
- Write the answer. Read the digits from the left, and put commas in.417,587
Check. Check: 268,860 + 417,587 = 686,447.
Watch out. When you exchange across zeros, change each zero to 9 as you pass it. Do not leave a zero in the middle.
Method. Subtract each column from the right. Exchange 1 from the column on the left when the top digit is too small.
- Subtract the ones column. 1 is less than 2, so exchange 1 tens for 10 ones.11 - 2 = 9
- Subtract the tens column. 3 is at least 3, so subtract directly.3 - 3 = 0
- Subtract the hundreds column. 8 is at least 8, so subtract directly.8 - 8 = 0
- Subtract the thousands column. 6 is at least 6, so subtract directly.6 - 6 = 0
- Subtract the ten thousands column. 4 is at least 4, so subtract directly.4 - 4 = 0
- Subtract the hundred thousands column. 7 is at least 6, so subtract directly.7 - 6 = 1
- Write the answer. Read the digits from the left, and put commas in.100,009
Check. Check: 646,832 + 100,009 = 746,841.
Watch out. When you exchange across zeros, change each zero to 9 as you pass it. Do not leave a zero in the middle.
Method. Subtract each column from the right. Exchange 1 from the column on the left when the top digit is too small.
- Subtract the ones column. 3 is less than 6, so exchange 1 tens for 10 ones.13 - 6 = 7
- Subtract the tens column. 3 is less than 5, so exchange 1 hundreds for 10 tens.13 - 5 = 8
- Subtract the hundreds column. 8 is at least 2, so subtract directly.8 - 2 = 6
- Subtract the thousands column. 5 is less than 9, so exchange 1 ten thousands for 10 thousands.15 - 9 = 6
- Subtract the ten thousands column. 4 is less than 5, so exchange 1 hundred thousands for 10 ten thousands.14 - 5 = 9
- Subtract the hundred thousands column. 7 is at least 4, so subtract directly.7 - 4 = 3
- Write the answer. Read the digits from the left, and put commas in.396,687
Check. Check: 459,256 + 396,687 = 855,943.
Watch out. When you exchange across zeros, change each zero to 9 as you pass it. Do not leave a zero in the middle.
Method. Subtract each column from the right. Exchange 1 from the column on the left when the top digit is too small.
- Subtract the ones column. 0 is less than 8, so exchange 1 tens for 10 ones.10 - 8 = 2
- Subtract the tens column. 6 is less than 8, so exchange 1 hundreds for 10 tens.16 - 8 = 8
- Subtract the hundreds column. 0 is less than 6, so exchange 1 thousands for 10 hundreds.10 - 6 = 4
- Subtract the thousands column. 5 is at least 2, so subtract directly.5 - 2 = 3
- Subtract the ten thousands column. 5 is at least 4, so subtract directly.5 - 4 = 1
- Subtract the hundred thousands column. 9 is at least 7, so subtract directly.9 - 7 = 2
- Write the answer. Read the digits from the left, and put commas in.213,482
Check. Check: 742,688 + 213,482 = 956,170.
Watch out. When you exchange across zeros, change each zero to 9 as you pass it. Do not leave a zero in the middle.
Method. Subtract each column from the right. Exchange 1 from the column on the left when the top digit is too small.
- Subtract the ones column. 1 is less than 9, so exchange 1 tens for 10 ones.11 - 9 = 2
- Subtract the tens column. 1 is less than 2, so exchange 1 hundreds for 10 tens.11 - 2 = 9
- Subtract the hundreds column. 4 is less than 9, so exchange 1 thousands for 10 hundreds.14 - 9 = 5
- Subtract the thousands column. 4 is less than 9, so exchange 1 ten thousands for 10 thousands.14 - 9 = 5
- Subtract the ten thousands column. 3 is less than 9, so exchange 1 hundred thousands for 10 ten thousands.13 - 9 = 4
- Subtract the hundred thousands column. 4 is at least 4, so subtract directly.4 - 4 = 0
- Write the answer. Read the digits from the left, and put commas in.45,592
Check. Check: 499,929 + 45,592 = 545,521.
Watch out. When you exchange across zeros, change each zero to 9 as you pass it. Do not leave a zero in the middle.
Estimating and checking
An estimate tells you roughly how big the answer should be. Round each number first, then calculate with the rounded numbers.
To check a sum, subtract one number from the total. To check a difference, add the number you took away back on.
Key ideas
- Round first, then calculate, to estimate.
- The exact answer should be close to the estimate.
- Check a sum by subtraction; check a difference by addition.
Method. Round each number first, then calculate with the rounded numbers.
- Round the first number. Look at the thousands digit: 5 or more rounds up.482,000 is about 480,000
- Round the second number. Use the same rule.310,000 is about 310,000
- Add the rounded numbers. This is the estimate.480,000 + 310,000 = 790,000
- Write the estimate. The exact answer should be close to this.790,000
Check. The exact answer is 792,000, which is close to 790,000.
Watch out. Estimate with the rounded numbers, then work out the exact answer with the column method.
Method. Check by using the inverse: subtract for a sum, or add back for a difference.
- Take one number away from the answer. If the sum is correct, taking one number away gives the other.700,001 - 582,463 = 117,538
- Compare with the other number. Is 117,538 the same as 117,538?Same
- Write Yes or No. Yes if the check works; No if it does not.Yes
Check. The correct answer is 700,001.
Watch out. A check that does not work means the answer is wrong. Find the mistake in the columns.
Method. Round each number first, then calculate with the rounded numbers.
- Round the first number. Look at the thousands digit: 5 or more rounds up.738,000 is about 740,000
- Round the second number. Use the same rule.205,000 is about 210,000
- Subtract the rounded numbers. This is the estimate.740,000 - 210,000 = 530,000
- Write the estimate. The exact answer should be close to this.530,000
Check. The exact answer is 533,000, which is close to 530,000.
Watch out. Estimate with the rounded numbers, then work out the exact answer with the column method.
Watch out
Watch out. An estimate is not the exact answer. Use the estimate to check that your answer is about the right size.
Exam tip
Grown-up tip. Ask the child: “Is the answer about the right size?” before they check it exactly.
Exercise 2D Practice: estimating and checking
Non-calculatorMethod. Check by using the inverse: subtract for a sum, or add back for a difference.
- Take one number away from the answer. If the sum is correct, taking one number away gives the other.688,072 - 569,853 = 118,219
- Compare with the other number. Is 118,219 the same as 118,219?Same
- Write Yes or No. Yes if the check works; No if it does not.Yes
Check. The correct answer is 688,072.
Watch out. A check that does not work means the answer is wrong. Find the mistake in the columns.
Method. Round each number first, then calculate with the rounded numbers.
- Round the first number. Look at the thousands digit: 5 or more rounds up.236,844 is about 240,000
- Round the second number. Use the same rule.314,998 is about 310,000
- Add the rounded numbers. This is the estimate.240,000 + 310,000 = 550,000
- Write the estimate. The exact answer should be close to this.550,000
Check. The exact answer is 551,842, which is close to 550,000.
Watch out. Estimate with the rounded numbers, then work out the exact answer with the column method.
Method. Check by using the inverse: subtract for a sum, or add back for a difference.
- Take one number away from the answer. If the sum is correct, taking one number away gives the other.994,651 - 854,966 = 139,685
- Compare with the other number. Is 139,685 the same as 139,685?Same
- Write Yes or No. Yes if the check works; No if it does not.Yes
Check. The correct answer is 994,651.
Watch out. A check that does not work means the answer is wrong. Find the mistake in the columns.
Method. Check by using the inverse: subtract for a sum, or add back for a difference.
- Take one number away from the answer. If the sum is correct, taking one number away gives the other.946,272 - 869,345 = 76,927
- Compare with the other number. Is 76,927 the same as 75,927?Not the same
- Write Yes or No. Yes if the check works; No if it does not.No
Check. The correct answer is 945,272.
Watch out. A check that does not work means the answer is wrong. Find the mistake in the columns.
Method. Round each number first, then calculate with the rounded numbers.
- Round the first number. Look at the thousands digit: 5 or more rounds up.448,428 is about 450,000
- Round the second number. Use the same rule.326,766 is about 330,000
- Add the rounded numbers. This is the estimate.450,000 + 330,000 = 780,000
- Write the estimate. The exact answer should be close to this.780,000
Check. The exact answer is 775,194, which is close to 780,000.
Watch out. Estimate with the rounded numbers, then work out the exact answer with the column method.
Method. Round each number first, then calculate with the rounded numbers.
- Round the first number. Look at the thousands digit: 5 or more rounds up.459,919 is about 460,000
- Round the second number. Use the same rule.25,866 is about 30,000
- Add the rounded numbers. This is the estimate.460,000 + 30,000 = 490,000
- Write the estimate. The exact answer should be close to this.490,000
Check. The exact answer is 485,785, which is close to 490,000.
Watch out. Estimate with the rounded numbers, then work out the exact answer with the column method.
Method. Check by using the inverse: subtract for a sum, or add back for a difference.
- Take one number away from the answer. If the sum is correct, taking one number away gives the other.304,704 - 228,141 = 76,563
- Compare with the other number. Is 76,563 the same as 76,573?Not the same
- Write Yes or No. Yes if the check works; No if it does not.No
Check. The correct answer is 304,714.
Watch out. A check that does not work means the answer is wrong. Find the mistake in the columns.
Method. Check by using the inverse: subtract for a sum, or add back for a difference.
- Take one number away from the answer. If the sum is correct, taking one number away gives the other.715,368 - 303,532 = 411,836
- Compare with the other number. Is 411,836 the same as 401,836?Not the same
- Write Yes or No. Yes if the check works; No if it does not.No
Check. The correct answer is 705,368.
Watch out. A check that does not work means the answer is wrong. Find the mistake in the columns.
Method. Check by using the inverse: subtract for a sum, or add back for a difference.
- Take one number away from the answer. If the sum is correct, taking one number away gives the other.930,992 - 834,723 = 96,269
- Compare with the other number. Is 96,269 the same as 106,269?Not the same
- Write Yes or No. Yes if the check works; No if it does not.No
Check. The correct answer is 940,992.
Watch out. A check that does not work means the answer is wrong. Find the mistake in the columns.
Method. Check by using the inverse: subtract for a sum, or add back for a difference.
- Take one number away from the answer. If the sum is correct, taking one number away gives the other.926,248 - 781,794 = 144,454
- Compare with the other number. Is 144,454 the same as 134,454?Not the same
- Write Yes or No. Yes if the check works; No if it does not.No
Check. The correct answer is 916,248.
Watch out. A check that does not work means the answer is wrong. Find the mistake in the columns.
Method. Check by using the inverse: subtract for a sum, or add back for a difference.
- Take one number away from the answer. If the sum is correct, taking one number away gives the other.869,153 - 266,137 = 603,016
- Compare with the other number. Is 603,016 the same as 603,016?Same
- Write Yes or No. Yes if the check works; No if it does not.Yes
Check. The correct answer is 869,153.
Watch out. A check that does not work means the answer is wrong. Find the mistake in the columns.
Method. Check by using the inverse: subtract for a sum, or add back for a difference.
- Take one number away from the answer. If the sum is correct, taking one number away gives the other.581,188 - 135,649 = 445,539
- Compare with the other number. Is 445,539 the same as 455,539?Not the same
- Write Yes or No. Yes if the check works; No if it does not.No
Check. The correct answer is 591,188.
Watch out. A check that does not work means the answer is wrong. Find the mistake in the columns.
Method. Check by using the inverse: subtract for a sum, or add back for a difference.
- Take one number away from the answer. If the sum is correct, taking one number away gives the other.867,377 - 345,902 = 521,475
- Compare with the other number. Is 521,475 the same as 521,475?Same
- Write Yes or No. Yes if the check works; No if it does not.Yes
Check. The correct answer is 867,377.
Watch out. A check that does not work means the answer is wrong. Find the mistake in the columns.
Method. Check by using the inverse: subtract for a sum, or add back for a difference.
- Take one number away from the answer. If the sum is correct, taking one number away gives the other.740,005 - 191,459 = 548,546
- Compare with the other number. Is 548,546 the same as 548,546?Same
- Write Yes or No. Yes if the check works; No if it does not.Yes
Check. The correct answer is 740,005.
Watch out. A check that does not work means the answer is wrong. Find the mistake in the columns.
Method. Round each number first, then calculate with the rounded numbers.
- Round the first number. Look at the thousands digit: 5 or more rounds up.392,244 is about 390,000
- Round the second number. Use the same rule.37,060 is about 40,000
- Subtract the rounded numbers. This is the estimate.390,000 - 40,000 = 350,000
- Write the estimate. The exact answer should be close to this.350,000
Check. The exact answer is 355,184, which is close to 350,000.
Watch out. Estimate with the rounded numbers, then work out the exact answer with the column method.
Method. Round each number first, then calculate with the rounded numbers.
- Round the first number. Look at the thousands digit: 5 or more rounds up.363,966 is about 360,000
- Round the second number. Use the same rule.191,116 is about 190,000
- Add the rounded numbers. This is the estimate.360,000 + 190,000 = 550,000
- Write the estimate. The exact answer should be close to this.550,000
Check. The exact answer is 555,082, which is close to 550,000.
Watch out. Estimate with the rounded numbers, then work out the exact answer with the column method.
Missing numbers and two-step problems
To find a missing number in a sum, subtract the known number from the total. To find a missing number in a difference, add.
In a two-step problem, work out the first step, then use the answer to work out the second step.
Key ideas
- Subtract to find a missing addend.
- Add to find a missing number in a difference.
- Do one step at a time, and check the answer fits the story.
Method. Use the inverse operation to find the missing number.
- Think about the total. The missing number is what is left when we take the known number away.300,000 - 248,000
- Subtract to find the missing number. Use the column method.300,000 - 248,000 = 52,000
- Check by adding. The missing number and the known number must make the total.248,000 + 52,000 = 300,000
- Write the missing number. This is the number in the box.52,000
Check. Check: put 52,000 into the sum and it works.
Watch out. To find a missing number in a sum, subtract. To find it in a difference, add.
Method. Work out the answer in the order the story happens, one step at a time.
- Work out the people left after some moved away. Take away the people who moved away.384,500 - 17,250 = 367,250
- Add the people who moved in. Add the new people to the number left.367,250 + 9,800 = 377,050
- Write the answer. Check the answer makes sense in the story.377,050
Check. Check: the answer should fit the story; the numbers left are less than the start unless people or things came in.
Watch out. Read the whole problem first. Decide the operation for each step before you calculate.
Method. Work out the answer in the order the story happens, one step at a time.
- Find the total picked. Add the two months together.412,000 + 287,500 = 699,500
- Take away the apples sold. Subtract the apples that were sold.699,500 - 150,000 = 549,500
- Write the answer. Check the answer makes sense in the story.549,500
Check. Check: the answer should fit the story; the numbers left are less than the start unless people or things came in.
Watch out. Read the whole problem first. Decide the operation for each step before you calculate.
Watch out
Watch out. In a two-step problem, the second step uses the answer of the first step, not the original number.
Exam tip
Grown-up tip. Ask the child to say the first step, and what it tells us, before they write any numbers.
Exercise 2E Practice: missing numbers and problems
Non-calculatorMethod. Work out the answer in the order the story happens, one step at a time.
- Find the total picked. Add the two months together.194,973 + 222,088 = 417,061
- Take away the apples sold. Subtract the apples that were sold.417,061 - 154,665 = 262,396
- Write the answer. Check the answer makes sense in the story.262,396
Check. Check: the answer should fit the story; the numbers left are less than the start unless people or things came in.
Watch out. Read the whole problem first. Decide the operation for each step before you calculate.
Method. Use the inverse operation to find the missing number.
- Think about the total. The missing number is what is left when we take the known number away.802,502 - 384,343
- Subtract to find the missing number. Use the column method.802,502 - 384,343 = 418,159
- Check by adding. The missing number and the known number must make the total.384,343 + 418,159 = 802,502
- Write the missing number. This is the number in the box.418,159
Check. Check: put 418,159 into the sum and it works.
Watch out. To find a missing number in a sum, subtract. To find it in a difference, add.
Method. Use the inverse operation to find the missing number.
- Think about the total. The missing number is what is left when we take the known number away.495,682 - 257,682
- Subtract to find the missing number. Use the column method.495,682 - 257,682 = 238,000
- Check by adding. The missing number and the known number must make the total.257,682 + 238,000 = 495,682
- Write the missing number. This is the number in the box.238,000
Check. Check: put 238,000 into the sum and it works.
Watch out. To find a missing number in a sum, subtract. To find it in a difference, add.
Method. Use the inverse operation to find the missing number.
- Add the number taken away to the answer. Adding undoes subtracting.83,088 + 37,426
- Work out the sum. Use the column method.83,088 + 37,426 = 120,514
- Check by subtracting. The answer must go back to the missing number.120,514 - 37,426 = 83,088
- Write the missing number. This is the number in the box.120,514
Check. Check: put 120,514 into the sum and it works.
Watch out. To find a missing number in a sum, subtract. To find it in a difference, add.
Method. Use the inverse operation to find the missing number.
- Think about the total. The missing number is what is left when we take the known number away.217,543 - 11,097
- Subtract to find the missing number. Use the column method.217,543 - 11,097 = 206,446
- Check by adding. The missing number and the known number must make the total.11,097 + 206,446 = 217,543
- Write the missing number. This is the number in the box.206,446
Check. Check: put 206,446 into the sum and it works.
Watch out. To find a missing number in a sum, subtract. To find it in a difference, add.
Method. Use the inverse operation to find the missing number.
- Think about the total. The missing number is what is left when we take the known number away.638,511 - 175,694
- Subtract to find the missing number. Use the column method.638,511 - 175,694 = 462,817
- Check by adding. The missing number and the known number must make the total.175,694 + 462,817 = 638,511
- Write the missing number. This is the number in the box.462,817
Check. Check: put 462,817 into the sum and it works.
Watch out. To find a missing number in a sum, subtract. To find it in a difference, add.
Method. Use the inverse operation to find the missing number.
- Think about the total. The missing number is what is left when we take the known number away.504,279 - 55,917
- Subtract to find the missing number. Use the column method.504,279 - 55,917 = 448,362
- Check by adding. The missing number and the known number must make the total.55,917 + 448,362 = 504,279
- Write the missing number. This is the number in the box.448,362
Check. Check: put 448,362 into the sum and it works.
Watch out. To find a missing number in a sum, subtract. To find it in a difference, add.
Method. Use the inverse operation to find the missing number.
- Add the number taken away to the answer. Adding undoes subtracting.249,478 + 110,200
- Work out the sum. Use the column method.249,478 + 110,200 = 359,678
- Check by subtracting. The answer must go back to the missing number.359,678 - 110,200 = 249,478
- Write the missing number. This is the number in the box.359,678
Check. Check: put 359,678 into the sum and it works.
Watch out. To find a missing number in a sum, subtract. To find it in a difference, add.
Method. Use the inverse operation to find the missing number.
- Think about the total. The missing number is what is left when we take the known number away.508,306 - 102,018
- Subtract to find the missing number. Use the column method.508,306 - 102,018 = 406,288
- Check by adding. The missing number and the known number must make the total.102,018 + 406,288 = 508,306
- Write the missing number. This is the number in the box.406,288
Check. Check: put 406,288 into the sum and it works.
Watch out. To find a missing number in a sum, subtract. To find it in a difference, add.
Method. Use the inverse operation to find the missing number.
- Think about the total. The missing number is what is left when we take the known number away.778,095 - 427,966
- Subtract to find the missing number. Use the column method.778,095 - 427,966 = 350,129
- Check by adding. The missing number and the known number must make the total.427,966 + 350,129 = 778,095
- Write the missing number. This is the number in the box.350,129
Check. Check: put 350,129 into the sum and it works.
Watch out. To find a missing number in a sum, subtract. To find it in a difference, add.
Method. Use the inverse operation to find the missing number.
- Add the number taken away to the answer. Adding undoes subtracting.340,542 + 281,988
- Work out the sum. Use the column method.340,542 + 281,988 = 622,530
- Check by subtracting. The answer must go back to the missing number.622,530 - 281,988 = 340,542
- Write the missing number. This is the number in the box.622,530
Check. Check: put 622,530 into the sum and it works.
Watch out. To find a missing number in a sum, subtract. To find it in a difference, add.
Method. Use the inverse operation to find the missing number.
- Add the number taken away to the answer. Adding undoes subtracting.346,409 + 78,701
- Work out the sum. Use the column method.346,409 + 78,701 = 425,110
- Check by subtracting. The answer must go back to the missing number.425,110 - 78,701 = 346,409
- Write the missing number. This is the number in the box.425,110
Check. Check: put 425,110 into the sum and it works.
Watch out. To find a missing number in a sum, subtract. To find it in a difference, add.
Method. Use the inverse operation to find the missing number.
- Think about the total. The missing number is what is left when we take the known number away.748,123 - 255,684
- Subtract to find the missing number. Use the column method.748,123 - 255,684 = 492,439
- Check by adding. The missing number and the known number must make the total.255,684 + 492,439 = 748,123
- Write the missing number. This is the number in the box.492,439
Check. Check: put 492,439 into the sum and it works.
Watch out. To find a missing number in a sum, subtract. To find it in a difference, add.
Method. Work out the answer in the order the story happens, one step at a time.
- Work out the people left after some moved away. Take away the people who moved away.499,278 - 193,221 = 306,057
- Add the people who moved in. Add the new people to the number left.306,057 + 81,406 = 387,463
- Write the answer. Check the answer makes sense in the story.387,463
Check. Check: the answer should fit the story; the numbers left are less than the start unless people or things came in.
Watch out. Read the whole problem first. Decide the operation for each step before you calculate.
Method. Use the inverse operation to find the missing number.
- Add the number taken away to the answer. Adding undoes subtracting.476,717 + 45,129
- Work out the sum. Use the column method.476,717 + 45,129 = 521,846
- Check by subtracting. The answer must go back to the missing number.521,846 - 45,129 = 476,717
- Write the missing number. This is the number in the box.521,846
Check. Check: put 521,846 into the sum and it works.
Watch out. To find a missing number in a sum, subtract. To find it in a difference, add.
Method. Use the inverse operation to find the missing number.
- Think about the total. The missing number is what is left when we take the known number away.486,234 - 141,505
- Subtract to find the missing number. Use the column method.486,234 - 141,505 = 344,729
- Check by adding. The missing number and the known number must make the total.141,505 + 344,729 = 486,234
- Write the missing number. This is the number in the box.344,729
Check. Check: put 344,729 into the sum and it works.
Watch out. To find a missing number in a sum, subtract. To find it in a difference, add.
Chapter summary
| Skill | Method |
|---|---|
| Column addition | Add from the right; carry the ten into the next column |
| Column subtraction | Exchange 1 from the left when the top digit is smaller; across zeros, make each zero a 9 |
| Estimating | Round each number to the nearest 10,000, then calculate |
| Checking | Subtract to check a sum; add back to check a difference |
| Missing numbers | Use the inverse: subtract to find an addend, add to find a number taken away |
| Two-step problems | Work out the first step, then use the answer for the next |
Can you…?
- I can add two numbers with up to six digits, carrying in several columns
- I can subtract in columns, exchanging across zeros
- I can estimate by rounding first
- I can check a sum by subtraction and a difference by addition
- I can find a missing number in a sum or difference
- I can solve two-step problems with numbers to a million
Key terms
- carry
- move a ten from one column to the column on its left
- exchange
- swap 1 from one column for 10 in the column on its right
- estimate
- a rough answer, found by rounding first
- inverse
- the operation that undoes another; subtraction undoes addition
Exercise 2Z Check-up
Non-calculatorMethod. Subtract each column from the right. Exchange 1 from the column on the left when the top digit is too small.
- Subtract the ones column. 8 is at least 3, so subtract directly.8 - 3 = 5
- Subtract the tens column. 6 is at least 2, so subtract directly.6 - 2 = 4
- Subtract the hundreds column. 2 is less than 4, so exchange 1 thousands for 10 hundreds.12 - 4 = 8
- Subtract the thousands column. 7 is at least 5, so subtract directly.7 - 5 = 2
- Subtract the ten thousands column. 3 is less than 9, so exchange 1 hundred thousands for 10 ten thousands.13 - 9 = 4
- Subtract the hundred thousands column. 0 is at least 0, so subtract directly.0 - 0 = 0
- Write the answer. Read the digits from the left, and put commas in.42,845
Check. Check: 95,423 + 42,845 = 138,268.
Watch out. When you exchange across zeros, change each zero to 9 as you pass it. Do not leave a zero in the middle.
Method. Use the inverse operation to find the missing number.
- Think about the total. The missing number is what is left when we take the known number away.604,641 - 258,175
- Subtract to find the missing number. Use the column method.604,641 - 258,175 = 346,466
- Check by adding. The missing number and the known number must make the total.258,175 + 346,466 = 604,641
- Write the missing number. This is the number in the box.346,466
Check. Check: put 346,466 into the sum and it works.
Watch out. To find a missing number in a sum, subtract. To find it in a difference, add.
Method. Check by using the inverse: subtract for a sum, or add back for a difference.
- Take one number away from the answer. If the sum is correct, taking one number away gives the other.575,432 - 442,296 = 133,136
- Compare with the other number. Is 133,136 the same as 133,146?Not the same
- Write Yes or No. Yes if the check works; No if it does not.No
Check. The correct answer is 575,442.
Watch out. A check that does not work means the answer is wrong. Find the mistake in the columns.
Method. Subtract each column from the right. Exchange 1 from the column on the left when the top digit is too small.
- Subtract the ones column. 5 is less than 9, so exchange 1 tens for 10 ones.15 - 9 = 6
- Subtract the tens column. 3 is less than 9, so exchange 1 hundreds for 10 tens.13 - 9 = 4
- Subtract the hundreds column. 6 is at least 3, so subtract directly.6 - 3 = 3
- Subtract the thousands column. 3 is less than 6, so exchange 1 ten thousands for 10 thousands.13 - 6 = 7
- Subtract the ten thousands column. 3 is at least 2, so subtract directly.3 - 2 = 1
- Subtract the hundred thousands column. 5 is at least 3, so subtract directly.5 - 3 = 2
- Write the answer. Read the digits from the left, and put commas in.217,346
Check. Check: 326,399 + 217,346 = 543,745.
Watch out. When you exchange across zeros, change each zero to 9 as you pass it. Do not leave a zero in the middle.
Method. Use the inverse operation to find the missing number.
- Add the number taken away to the answer. Adding undoes subtracting.338,690 + 211,998
- Work out the sum. Use the column method.338,690 + 211,998 = 550,688
- Check by subtracting. The answer must go back to the missing number.550,688 - 211,998 = 338,690
- Write the missing number. This is the number in the box.550,688
Check. Check: put 550,688 into the sum and it works.
Watch out. To find a missing number in a sum, subtract. To find it in a difference, add.
Method. Use the inverse operation to find the missing number.
- Think about the total. The missing number is what is left when we take the known number away.409,082 - 102,640
- Subtract to find the missing number. Use the column method.409,082 - 102,640 = 306,442
- Check by adding. The missing number and the known number must make the total.102,640 + 306,442 = 409,082
- Write the missing number. This is the number in the box.306,442
Check. Check: put 306,442 into the sum and it works.
Watch out. To find a missing number in a sum, subtract. To find it in a difference, add.
Method. Add the columns from the right. Carry any ten into the next column on the left.
- Add the ones column. Write 7 in the ones column.1 + 6 = 7
- Add the tens column. Write 3 and carry 1 to the hundreds column.5 + 8 = 13
- Add the hundreds column. Write 9 and carry 1 to the thousands column.9 + 9 + 1 carried = 19
- Add the thousands column. Write 0 and carry 1 to the ten thousands column.9 + 0 + 1 carried = 10
- Add the ten thousands column. Write 4 and carry 1 to the hundred thousands column.8 + 5 + 1 carried = 14
- Add the hundred thousands column. Write 9 in the hundred thousands column.8 + 0 + 1 carried = 9
- Write the answer. Put the digits together, with commas between groups of three.940,937
Check. Check: 940,937 - 50,986 = 889,951.
Watch out. Carry the 1 into the next column. Do not forget it, and do not write it as a digit in its own column.
Method. Subtract each column from the right. Exchange 1 from the column on the left when the top digit is too small.
- Subtract the ones column. 8 is at least 5, so subtract directly.8 - 5 = 3
- Subtract the tens column. 7 is less than 9, so exchange 1 hundreds for 10 tens.17 - 9 = 8
- Subtract the hundreds column. 3 is less than 8, so exchange 1 thousands for 10 hundreds.13 - 8 = 5
- Subtract the thousands column. 7 is less than 8, so exchange 1 ten thousands for 10 thousands.17 - 8 = 9
- Subtract the ten thousands column. 5 is at least 3, so subtract directly.5 - 3 = 2
- Subtract the hundred thousands column. 4 is at least 0, so subtract directly.4 - 0 = 4
- Write the answer. Read the digits from the left, and put commas in.429,583
Check. Check: 38,895 + 429,583 = 468,478.
Watch out. When you exchange across zeros, change each zero to 9 as you pass it. Do not leave a zero in the middle.
Method. Add the columns from the right. Carry any ten into the next column on the left.
- Add the ones column. Write 0 and carry 1 to the tens column.7 + 3 = 10
- Add the tens column. Write 5 in the tens column.2 + 2 + 1 carried = 5
- Add the hundreds column. Write 3 and carry 1 to the thousands column.4 + 9 = 13
- Add the thousands column. Write 7 in the thousands column.4 + 2 + 1 carried = 7
- Add the ten thousands column. Write 9 in the ten thousands column.9 + 0 = 9
- Add the hundred thousands column. Write 4 in the hundred thousands column.0 + 4 = 4
- Write the answer. Put the digits together, with commas between groups of three.497,350
Check. Check: 497,350 - 402,923 = 94,427.
Watch out. Carry the 1 into the next column. Do not forget it, and do not write it as a digit in its own column.
Method. Subtract each column from the right. Exchange 1 from the column on the left when the top digit is too small.
- Subtract the ones column. 3 is less than 5, so exchange 1 tens for 10 ones.13 - 5 = 8
- Subtract the tens column. 2 is at least 2, so subtract directly.2 - 2 = 0
- Subtract the hundreds column. 2 is less than 6, so exchange 1 thousands for 10 hundreds.12 - 6 = 6
- Subtract the thousands column. 1 is at least 1, so subtract directly.1 - 1 = 0
- Subtract the ten thousands column. 7 is at least 4, so subtract directly.7 - 4 = 3
- Subtract the hundred thousands column. 7 is at least 1, so subtract directly.7 - 1 = 6
- Write the answer. Read the digits from the left, and put commas in.630,608
Check. Check: 141,625 + 630,608 = 772,233.
Watch out. When you exchange across zeros, change each zero to 9 as you pass it. Do not leave a zero in the middle.
Total 19 marks. Allow about 19 minutes.
Exercise 2X Practice test questions
Non-calculatorMethod. Round each number first, then calculate with the rounded numbers.
- Round the first number. Look at the thousands digit: 5 or more rounds up.249,766 is about 250,000
- Round the second number. Use the same rule.351,138 is about 350,000
- Add the rounded numbers. This is the estimate.250,000 + 350,000 = 600,000
- Write the estimate. The exact answer should be close to this.600,000
Check. The exact answer is 600,904, which is close to 600,000.
Watch out. Estimate with the rounded numbers, then work out the exact answer with the column method.
Method. Add the columns from the right. Carry any ten into the next column on the left.
- Add the ones column. Write 6 in the ones column.5 + 1 = 6
- Add the tens column. Write 1 and carry 1 to the hundreds column.5 + 6 = 11
- Add the hundreds column. Write 8 in the hundreds column.4 + 3 + 1 carried = 8
- Add the thousands column. Write 9 in the thousands column.6 + 3 = 9
- Add the ten thousands column. Write 4 and carry 1 to the hundred thousands column.7 + 7 = 14
- Add the hundred thousands column. Write 9 in the hundred thousands column.5 + 3 + 1 carried = 9
- Write the answer. Put the digits together, with commas between groups of three.949,816
Check. Check: 949,816 - 373,361 = 576,455.
Watch out. Carry the 1 into the next column. Do not forget it, and do not write it as a digit in its own column.
Method. Subtract each column from the right. Exchange 1 from the column on the left when the top digit is too small.
- Subtract the ones column. 1 is at least 1, so subtract directly.1 - 1 = 0
- Subtract the tens column. 4 is at least 2, so subtract directly.4 - 2 = 2
- Subtract the hundreds column. 4 is at least 0, so subtract directly.4 - 0 = 4
- Subtract the thousands column. 7 is less than 9, so exchange 1 ten thousands for 10 thousands.17 - 9 = 8
- Subtract the ten thousands column. 0 is less than 1, so exchange 1 hundred thousands for 10 ten thousands.10 - 1 = 9
- Subtract the hundred thousands column. 7 is at least 1, so subtract directly.7 - 1 = 6
- Write the answer. Read the digits from the left, and put commas in.698,420
Check. Check: 119,021 + 698,420 = 817,441.
Watch out. When you exchange across zeros, change each zero to 9 as you pass it. Do not leave a zero in the middle.
Method. Add the columns from the right. Carry any ten into the next column on the left.
- Add the ones column. Write 9 in the ones column.5 + 4 = 9
- Add the tens column. Write 6 in the tens column.3 + 3 = 6
- Add the hundreds column. Write 3 and carry 1 to the thousands column.4 + 9 = 13
- Add the thousands column. Write 0 and carry 1 to the ten thousands column.3 + 6 + 1 carried = 10
- Add the ten thousands column. Write 8 in the ten thousands column.7 + 0 + 1 carried = 8
- Add the hundred thousands column. Write 9 in the hundred thousands column.8 + 1 = 9
- Write the answer. Put the digits together, with commas between groups of three.980,369
Check. Check: 980,369 - 106,934 = 873,435.
Watch out. Carry the 1 into the next column. Do not forget it, and do not write it as a digit in its own column.
Method. Add the columns from the right. Carry any ten into the next column on the left.
- Add the ones column. Write 7 in the ones column.6 + 1 = 7
- Add the tens column. Write 7 and carry 1 to the hundreds column.9 + 8 = 17
- Add the hundreds column. Write 4 in the hundreds column.2 + 1 + 1 carried = 4
- Add the thousands column. Write 2 and carry 1 to the ten thousands column.8 + 4 = 12
- Add the ten thousands column. Write 8 in the ten thousands column.5 + 2 + 1 carried = 8
- Add the hundred thousands column. Write 5 in the hundred thousands column.4 + 1 = 5
- Write the answer. Put the digits together, with commas between groups of three.582,477
Check. Check: 582,477 - 124,181 = 458,296.
Watch out. Carry the 1 into the next column. Do not forget it, and do not write it as a digit in its own column.
Method. Subtract each column from the right. Exchange 1 from the column on the left when the top digit is too small.
- Subtract the ones column. 4 is less than 7, so exchange 1 tens for 10 ones.14 - 7 = 7
- Subtract the tens column. 1 is less than 3, so exchange 1 hundreds for 10 tens.11 - 3 = 8
- Subtract the hundreds column. 7 is at least 2, so subtract directly.7 - 2 = 5
- Subtract the thousands column. 9 is at least 5, so subtract directly.9 - 5 = 4
- Subtract the ten thousands column. 8 is at least 4, so subtract directly.8 - 4 = 4
- Subtract the hundred thousands column. 1 is at least 0, so subtract directly.1 - 0 = 1
- Write the answer. Read the digits from the left, and put commas in.144,587
Check. Check: 45,237 + 144,587 = 189,824.
Watch out. When you exchange across zeros, change each zero to 9 as you pass it. Do not leave a zero in the middle.
Method. Subtract each column from the right. Exchange 1 from the column on the left when the top digit is too small.
- Subtract the ones column. 4 is less than 8, so exchange 1 tens for 10 ones.14 - 8 = 6
- Subtract the tens column. 1 is less than 3, so exchange 1 hundreds for 10 tens.11 - 3 = 8
- Subtract the hundreds column. 3 is less than 4, so exchange 1 thousands for 10 hundreds.13 - 4 = 9
- Subtract the thousands column. 5 is at least 0, so subtract directly.5 - 0 = 5
- Subtract the ten thousands column. 7 is at least 3, so subtract directly.7 - 3 = 4
- Subtract the hundred thousands column. 9 is at least 3, so subtract directly.9 - 3 = 6
- Write the answer. Read the digits from the left, and put commas in.645,986
Check. Check: 330,438 + 645,986 = 976,424.
Watch out. When you exchange across zeros, change each zero to 9 as you pass it. Do not leave a zero in the middle.
Method. Check by using the inverse: subtract for a sum, or add back for a difference.
- Take one number away from the answer. If the sum is correct, taking one number away gives the other.845,615 - 633,340 = 212,275
- Compare with the other number. Is 212,275 the same as 211,275?Not the same
- Write Yes or No. Yes if the check works; No if it does not.No
Check. The correct answer is 844,615.
Watch out. A check that does not work means the answer is wrong. Find the mistake in the columns.
Method. Use the inverse operation to find the missing number.
- Think about the total. The missing number is what is left when we take the known number away.494,697 - 460,804
- Subtract to find the missing number. Use the column method.494,697 - 460,804 = 33,893
- Check by adding. The missing number and the known number must make the total.460,804 + 33,893 = 494,697
- Write the missing number. This is the number in the box.33,893
Check. Check: put 33,893 into the sum and it works.
Watch out. To find a missing number in a sum, subtract. To find it in a difference, add.
Method. Use the inverse operation to find the missing number.
- Think about the total. The missing number is what is left when we take the known number away.539,434 - 351,316
- Subtract to find the missing number. Use the column method.539,434 - 351,316 = 188,118
- Check by adding. The missing number and the known number must make the total.351,316 + 188,118 = 539,434
- Write the missing number. This is the number in the box.188,118
Check. Check: put 188,118 into the sum and it works.
Watch out. To find a missing number in a sum, subtract. To find it in a difference, add.